It doesn’t take too much delving around with the primes to come across the appearance of Euler’s totient or phi function. It oozes out of the primes in a variety of locations.

Firstly the composite helices and thus the prime helices each have a “width”. The width of the 6n+/-1 helix associated with prime factor 5 is 2. The width of the next helix associated with prime factor 7 is 8. The next is 48 and the conjecture is that this continues:

480

5760

etc

Secondly, considering the balance of primes to composites as each prime factor is used to calculate which are composite and which are potentially prime we see a simple pattern:

Each block is p_{i}# integers long,

Each block has (p_{i } – 1) # (potential) primes & (p_{i-1 }-1)# composites

Thus the ratio of

primes to all integers: (p_{i } – 1) # / p_{i}#

primes to composites (p_{i } – 1) # / (p_{i-1 }-1) # = p_{i } – 1